Search for "Alan Shepard Park"
Named after the astronaut. It has a large parking lot. $5 a day to park. There are several vendors on the beach for chairs and umbrellas and such. Very clean, about a mile south of the pier.
To OP.....
Also check out Lori Wilson Park. Has parking, restrooms, dog park(not on the beach itself), nature trail, playground. It doesn't always have food, but weekends sometimes there is a hot dog type stand with Italian Ice etc. , It is not far from Ron Jons and the other tourist surf shop. Great picture ops at Ron Jons with the statues and surf wagon.
As for Clearwater or Cocoa Beach being better......neither is better, they are just different experience.
Clearwater is shallow water farther out with hardly any wave action in a Gulf. So kids can sit 40 yards from shore at times and play in the water.
Cocoa Beach is on the Atlantic Ocean so it has a little more wave action. The bottom there is still pretty level compared to farther South in Brevard county so we generally take our younger nieces and nephews to Lori Wilson instead of here in Melbourne unless they are adventurous and want to boogy board or try to surf. You will get to watch surfers on the East coast.
Another thought......check out Sebastian Inlet for a fun day......
True the sands on the Gulf side are white but the sands on the ocean side are a special sand that some types of sea turtles will only return to to nest. We don't mess with the sand. Even when hurricanes destroy our beaches we only refill with the natural sand from the bottom of the ocean rather than trucking in sand from elsewhere.
As for finding a lot of shells on either beach....not many in Cocoa Beach unless waters rough from storms etc. Don't remember any shells on Clearwater either. Need to go farther South on both sides to pick up better shells usually. South down by Melbourne has more shells on East Coast during the late summer and if they have not done a lot of dredging recently to try to save the dunes. Have not seen many shells on beach this year yet.